Affiliations: Dept. of Computer Science, Stevens Institute of
Technology, Hoboken, NJ, USA | GRLMC, Rovira i Virgili University, Tarragona, Spain.
E-mail: ze@inf.u-szeged.hu | Inst. for Discrete Mathematics and Geometry, Technical
University of Vienna, Austria
Note: [] Address for correspondence: GRLMC, Rovira i Virgili University,
Tarragona, Spain
Abstract: A Conway semiring is a semiring S equipped with a unary operation *:
S → S, always called 'star', satisfying the sum star and product star
identities. It is known that these identities imply a Kleene type theorem. Some
computationally important semirings, such as N or
N^{rat}≪Σ≫ of rational power series of words on
Σ with coefficients in N, cannot have a total star operation satisfying
the Conway identities. We introduce here partial Conway semirings, which are
semirings S which have a star operation defined only on an ideal of S; when the
arguments are appropriate, the operation satisfies the above identities. We
develop the general theory of partial Conway semirings and prove a Kleene
theorem for this generalization.
